Antisymmetrical tensor.
A contravariant or co-variant tensor of the 2nd, 3rd or
4th rank is called antisymmetrical when the two components
got by mutually interchanging any two indices
are equal and opposite. The tensor or A^{μν} or A_{μν} is thus
antisymmetrical when we have
(15) A^{μν} = -A^{νμ}
or
(15a) A_{μν} = -A_{νμ}.
Of the 16 components A^{μν}, the four components A^{μμ} vanish, the rest are equal and opposite in pairs; so that there are only 6 numerically different components present (Six-vector).
Thus we also see that the antisymmetrical tensor A^{μνσ} (3rd rank) has only 4 components numerically different, and the antisymmetrical tensor A^{μνστ} only one. Symmetrical tensors of ranks higher than the fourth, do not exist in a continuum of 4 dimensions.
§ 7. Multiplication of Tensors.
Outer multiplication of Tensors:—We get from the components of a tensor of rank z, and another of a rank z´, the components of a tensor of rank (z + z´) for which we multiply all the components of the first with all the components of the second in pairs. For example, we
